The inverse Galois problem for p-adic fields

We now apply the cohomology theory of the previous chapter to the study of the Brauer group. However, we shall have to use a slightly modified construction which takes into account the fact that the absolute Galois group of a field is determined by its finite quotients. This is the cohomology theory of profinite groups, which we develop first. As a fruit of our labours, we identify the Brauer group of a field with a second, this time commutative, cohomology group of the absolute Galois group. As applications, we obtain easy proofs of some basic facts concerning the Brauer group, and give an important characterization of the index of a central simple algebra. Last but not least, one of the main objects of study in this book makes its appearance: the Galois symbol. The cohomology theory of profinite groups was introduced in the late 1950s by John Tate, motivated by sheaf-theoretic considerations of Alexander Grothendieck. His original aim was to find the appropriate formalism for developing class field theory. Tate himself never published his work, which thus became accessible to the larger mathematical community through the famous account of Serre [4], which also contains many original contributions. It was Brauer himself who described the Brauer group as a second cohomology group, using his language of factor systems. We owe to Serre the insight that descent theory can be used to give a more conceptual proof. The Galois symbol was defined by Tate in connection with the algebraic theory of power residue symbols, a topic extensively studied in the 1960s by Bass, Milnor, Moore, Serre and others. Profinite groups and Galois groups It can be no surprise that the main application of the cohomological techniques of the previous chapter will be in the case when G is the Galois group of a finite Galois extension. However, it will be convenient to consider the case of infinite Galois extensions as well, and first and foremost that of the extension k s |k , where k s is a separable closure of k .

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups H. We provide a unified study of the action of both cover Galois group H and the absolute Galois group $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely describe the wild fundamental group and apply this result to the inverse problem in $q$-difference Galois theory. The new version contains an appendix on pronilpotent completion and the main result on the direct problem is made more precise. (Submitted for publication)

Let K be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [3]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFAesuaaa!40A2! $$\mathbb{Q}$$ whose Galois group is 2-elementary abelian. Then we shall prove that F of degree % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaadA % eacaGG6aWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaa % cqWFAesucaGGDbGaeyyzImRaaGioaaaa!4673! $$[F:\mathbb{Q}] \geqq 8$$ , is monogenic if and only if % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2 % da9mrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8Ng % HeLaaiikamaakaaabaGaeyOeI0IaaGymaaWcbeaakiaacYcadaGcaa % qaaiabgkHiTiaaikdaaSqabaGccaGGSaWaaOaaaeaacqGHsislcaaI % ZaaaleqaaOGaaiykaiabg2da9iab-PrirjaacIcacqaH2oGEdaWgaa % WcbaGaaGOmaiaaisdaaeqaaOGaaiykaaaa!5173! $$F = \mathbb{Q}(\sqrt { - 1} ,\sqrt { - 2} ,\sqrt { - 3} ) = \mathbb{Q}(\zeta _{24} )$$ under a suitable condition for the case of degree 8.

Galois groups of global fields of finite characteristic

A characterization of number fields by p-closed extensions

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

In this last chapter we want to give some idea of the “anabelian” program of A. Grothendieck. The term anabelian should be read as “far from being abelian” and as we understand the matter, a group is far enough away from being abelian if all of its subgroups of finite index have a trivial center. The principal idea is the following: in topology, a space X of type K(π,1) is determined by its fundamental group π up to weak homotopy equivalence. If we require that X is a CW-complex, then X is already determined up to strong homotopy equivalence. The “anabelian” idea is that something similar should also be true for schemes, i.e. a scheme X which is an étale K(π,1) should be essentially reconstructible from its étale fundamental group. This is obviously not correct in general, but it should be true under certain conditions; for example, X should be absolutely finitely generated and $\pi^{\mathit{et}}_{1}(X)$ is supposed to be “anabelian”. The smallest constituents of this anabelian world are points, i.e. spectra of fields which are finitely generated over their prime fields. Here the étale fundamental group is just the absolute Galois group. Finite fields have an abelian absolute Galois group (in fact all these fields have the same Galois group $\hat{\mathbb{Z}}$ ), and so the first objects of interest are global fields. In §1,2 we will present some results on “anabelian properties” of global fields, which already existed before Grothendieck formulated his program. We will explain the general conjectures in §3.

A celebrated theorem of Artin and Schreier [AS] characterizes the fields K whose absolute Galois group GK is isomorphic to that of R as the real closed fields. In the present paper we consider the analogous problem for nonarchimedean local fields of positive characteristic F = Fpn((t)). We show that a field K with absolute Galois group isomorphic to GF possesses a Henselian valuation v such that: (1) the value group Γ of v satisfies Γ/l ∼= Z/l for all prime numbers l = p; (2) the residue field K of v has characteristic p; (3) the maximal prime to p Galois group GK(p) of K is Ẑ/Zp; (4) if charK = 0 then Γ = pΓ and K is perfect. For every positive integer r we construct such fields K of characteristic p with Γ/p ∼= (Z/p). Likewise we construct examples with Γ ∼= Z, GK ∼= Ẑ and K imperfect. The similar problem for p-adic fields was answered by Koenigsmann [Kn] and the first named author [E1] (for p = 2), extending earlier results by Neukirch [N2] and Pop [P1]: the fields K such that GK ∼= GF for some finite extension F of Qp are precisely the p-adically closed fields in the sense of [PR].

We calculate the number of Galois extensions, up to isomorphism, of a local field whose Galois groups are isomorphic to the semidihedral (resp. modular) group of order $2^m$ ($m\ge 4$).

A real matrix A is said to be an M-matrix if it is of the form where B is a nonnegative matrix with Perron eigenvalue and . This paper provides sufficient conditions for the existence and construction of an M-matrix A with prescribed elementary divisors, which are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A. This inverse problem on M-matrices has not been treated until now. We solve the inverse elementary divisors problem for diagonalizable M-matrices and the symmetric generalized doubly stochastic inverse M-matrix problem for lists of real numbers and for lists of complex numbers of the form . The constructive nature of our results allows for the computation of a solution matrix. The paper also discusses an application of M-matrices to a capacity problem in wireless communications.

The unique solvability of linear inverse coefficient problems for the evolutionary equation in a Banach space with the Caputo Fabrizio derivative is studied. An operator at the unknown function in the equation is assumed to be bounded, the equation is endowed with the Cauchy condition. For the inverse problem with a constant unknown coefficient and with an integral overdefinition condition in the sense of Riemann Stieltjes, which includes the condition of final overdefinition as a special case, a well-posedness criterion is obtained. Sufficient conditions for unique solvability and an estimate of the well-posedness for the solution are obtained for a linear inverse problem with a time-dependent unknown coefficient. The abstract results obtained are used in the study of inverse problems with an unknown coefficient depending only on spatial variables or only on time, for equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

On inverse problem for a convolution integro-differential operator with Robin boundary conditions

We study the distribution of the Galois group of a random q q -additive polynomial over a rational function field: For q q a power of a prime p p , let f = X q n + a n − 1 X q n − 1 + … + a 1 X q + a 0 X f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X be a random polynomial chosen uniformly from the set of q q -additive polynomials of degree n n and height d d , that is, the coefficients are independent uniform polynomials of degree deg ⁡ a i ≤ d \deg a_i\leq d . The Galois group G f G_f is a random subgroup of GL n ⁡ ( q ) \operatorname {GL}_n(q) . Our main result shows that G f G_f is almost surely large as d , q d,q are fixed and n → ∞ n\to \infty . For example, we give necessary and sufficient conditions so that SL n ⁡ ( q ) ≤ G f \operatorname {SL}_n(q)\leq G_f asymptotically almost surely. Our proof uses the classification of maximal subgroups of GL n ⁡ ( q ) \operatorname {GL}_n(q) . We also consider the limits: q , n q,n fixed, d → ∞ d\to \infty and d , n d,n fixed, q → ∞ q\to \infty , which are more elementary.